AUSTRIAN-POLISH 1995

Determine all real solutions $(a_1,...,a_n)$ of the following system of equations: $$\begin{cases}a_3 = a_2 + a_1 a_4 = a_3 + a_2 ... a_n = a_{n-1} + a_{n-2} a_1= a_n +a_{n-1} a_2 = a_1 + a_n \end{cases}$$

Let $X= \{A_1, A_2, A_3, A_4\}$ be a set of four distinct points in the plane. Show that there exists a subset $Y$ of $X$ with the property that there is no (closed) disk $K$ such that $K\cap X = Y$ .

Let $P(x) = x^4 + x^3 + x^2 + x + 1$ . Show that there exist two non-constant polynomials $Q(y)$ and $R(y)$ with integer coefficients such that for all $Q(y) \cdot R(y)= P(5y^2)$ for all $y$ .

Determine all polynomials $P(x)$ with real coefficients such that $P(x)^2 + P\left(\frac{1}{x}\right)^2= P(x^2)P\left(\frac{1}{x^2}\right)$ for all $x$ .

$ABC$ is an equilateral triangle. $A_{1}, B_{1}, C_{1}$ are the midpoints of $BC, CA, AB$ respectively. $p$ is an arbitrary line through $A_{1}$ . $q$ and $r$ are lines parallel to $p$ through $B_{1}$ and $C_{1}$ respectively. $p$ meets the line $B_{1}C_{1}$ at $A_{2}$ . Similarly, $q$ meets $C_{1}A_{1}$ at $B_{2}$ , and $r$ meets $A_{1}B_{1}$ at $C_{2}$ . Show that the lines $AA_{2}, BB_{2}, CC_{2}$ meet at some point $X$ , and that $X$ lies on the circumcircle of $ABC$ .

The Alpine Club organizes four mountain trips for its $n$ members. Let $E_1, E_2, E_3, E_4$ be the teams participating in these trips. In how many ways can these teams be formed so as to satisfy $E_1 \cap E_2 \ne\varnothing$ , $E_2 \cap E_3 \ne\varnothing$ , $E_3 \cap E_4 \ne\varnothing$ ?

Consider the equation $3y^4 + 4cy^3 + 2xy + 48 = 0$ , where $c$ is an integer parameter. Determine all values of $c$ for which the number of integral solutions $(x,y)$ satisfying the conditions (i) and (ii) is maximal: (i) $|x|$ is a square of an integer; (ii) $y$ is a squarefree number.

Consider the cube with the vertices at the points $(\pm 1, \pm 1, \pm 1)$ . Let $V_1,...,V_{95}$ be arbitrary points within this cube. Denote $v_i = \overrightarrow{OV_i}$ , where $O = (0,0,0)$ is the origin. Consider the $2^{95}$ vectors of the form $s_1v_1 + s_2v_2 +...+ s_{95}v_{95}$ , where $s_i = \pm 1$ . (a) If $d = 48$ , prove that among these vectors there is a vector $w = (a, b, c)$ such that $a^2 + b^2 + c^2 \le 48$ . (b) Find a smaller $d$ (the smaller, the better) with the same property.

Prove that for all positive integers $n,m$ and all real numbers $x, y > 0$ the following inequality holds: $$(n - 1)(m- 1)(x^{n+m} + y^{n+m}) + (n + m - 1)(x^ny^m + x^my^n) \ge nm(x^{n+m-1}y + xy^{n+m-1}).$$